One of the most important issues in the field of ranking voting systems is the choice of the weighting vector. This issue has been addressed in the literature from different approaches, and one of them has been to obtain the weighting vector as a solution to a linear programming problem. In this paper we analyze some models proposed in the literature and show that one of their main shortcomings is that they cannot guarantee the uniqueness of the solution, so the winner or the final ranking of the candidates may depend on the chosen weighting vector. An alternative to these models is the use of surrogate weights, among which rank order centroid (ROC) weights stand out as the centroid of a specific simplex. Following this idea, in this paper we show the explicit expression for the weights that form the centroid of diverse simplices utilized in ranking voting systems, and we also see that certain surrogate weights frequently employed in literature can be derived as extreme cases where the simplices collapse into a single vector. Moreover, we argue that averaging two weighting vectors can be a valid approach in some cases and, in this way, we can get weighting vectors that closely resemble those used in some sports competitions.

中文翻译：

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排名投票系统和代理权重：质心权重的显式公式

排名投票系统领域最重要的问题之一是权重向量的选择。文献中已经通过不同的方法解决了这个问题，其中之一是获得权重向量作为线性规划问题的解决方案。在本文中，我们分析了文献中提出的一些模型，并表明它们的主要缺点之一是无法保证解决方案的唯一性，因此获胜者或候选者的最终排名可能取决于所选的权重向量。这些模型的替代方法是使用代理权重，其中排序质心 (ROC) 权重作为特定单纯形的质心脱颖而出。按照这个想法，在本文中，我们展示了形成排名投票系统中使用的不同单纯形质心的权重的显式表达式，并且我们还看到文献中经常使用的某些替代权重可以导出为单纯形崩溃的极端情况转化为单个向量。此外，我们认为在某些情况下对两个权重向量进行平均可能是一种有效的方法，通过这种方式，我们可以获得与某些体育比赛中使用的权重向量非常相似的权重向量。